After spending hours trying to get the thing working again, I found the solution.
For me, it was incomprehensible and also incomprehensible. I hard reset the modem, which is connected to the fiber optic line and distributes the Internet to the devices via Lan --> Switch. Now you can add new devices to all SSIDs again.

It's great that it works again, but I would like to understand why a modem that only loops the signal through is responsible for not being able to register new devices in the network? Absolutely no idea...